Post on 27-May-2015
Unit 1 Functions and Relations
1-1 Number Theory Number Systems Rational and Irrational Numbers
1-2 Functions and Linear Graphs Functions and Function Notation 1-1 and Onto Graphing
1-3 Equations and Inequalities Solving Linear and Quadratic Equations and Inequalities Solving for a Variable
1-1 Number Theory
Unit 1 Functions and Relations
Concepts and Objectives
Number Theory (Obj. #1) Identify subsets of real numbers Simplify expressions using order of operations Identify real number axioms
Rational Numbers (Obj. #2) Convert between fractions and decimals
Number Systems
What we currently know as the set of real numbers was only formulated around 1879. We usually present this as sets of numbers.
Number Systems
The set of natural numbers () and the set of integers () have been around since ancient times, probably prompted by the need to maintain trade accounts. Ancient civilizations, such as the Babylonians, also used ratios to compare quantities.
One of the greatest mathematical advances was the introduction of the number 0.
Properties of Real Numbers
Closure Property a + b ab
Commutative Property a + b = b + a ab = ba
Associative Property (a + b) + c = a + (b + c) (ab)c = a(bc)
Identity Property a + 0 = a a 1 = a
Inverse Property a + (–a) = 0
Distributive Property a(b + c) = ab + ac
For all real numbers a, b, and c:
1
=1 aa
Properties of Real Numbers
The properties are also called axioms. 0 is called the additive identity and 1 is called the
multiplicative identity. Notice the relationships between the identities and the
inverses (called the additive inverse and the multiplicative inverse).
Saying that a set is “closed” under an operation (such as multiplication) means that performing that operation on numbers in the set will always produce an answer that is also in the set – there are no answers outside the set.
Properties of Real Numbers
Examples The set of natural numbers () is not closed under the
operation of subtraction. Why?
–20 5 = –4. Does this show that the set of integers is closed under division?
Properties of Real Numbers
Examples The set of natural numbers () is not closed under the
operation of subtraction. Why? 5 – 7 = –2, which is not in .
–20 5 = –4. Does this show that the set of integers is closed under division?
No. Any division that has a remainder is not in .
Order of Operations
Parentheses (or other grouping symbols, such as square brackets or fraction bars) – start with the innermost set, following the sequence below, and work outward.
Exponents Multiplication Division Addition Subtraction
working from left to right
working from left to right
Order of Operations
Use order of operations to explain why
We can think of –3 as being –1 3. Therefore we have
It should be easier now to see that on the left side we multiply first and then apply the exponent, and on the right side, we apply the exponent and then multiply.
2 23 3
2 21 3 1 3
Order of Operations
Work the following examples without using your calculator.
1.
2.
3.
2 5 12 3
34 9 8 7 2
8 4 6 12
4 3
Order of Operations
Work the following examples without using your calculator.
1.
2.
3.
2 5 12 3
34 9 8 7 2
8 4 6 12
4 3
1. –6
2. –60
6
3. 7
Absolute Value
The absolute value of a real number a, denoted by |a|, is the distance from 0 to a on the number line. This distance is always taken to be nonnegative.
if 0
if 0
x xx
x x
Absolute Value Properties
For all real numbers a and b:
1.
2.
3.
4.
5.
0a
a a
a b ab
( 0)a a
bb b
a b a b
Absolute Value
Example: Rewrite each expression without absolute value bars.
1.
2.
3.
3 1
2
, if 0x
xx
Absolute Value
Example: Rewrite each expression without absolute value bars.
1.
2.
3.
3 1
2
, if 0x
xx
1. 3 1
2. – 2
3. –1
Rational Numbers
The Greeks, specifically Pythagoras of Samos, originally believed that the lengths of all segments in geometric objects could be expressed as ratios of positive integers.
A number is a rational number () if and only if it can be expressed as the ratio (or quotient) of two integers.
Rational numbers include decimals as well as fractions. The definition does not require that a rational number must be written as a quotient of two integers, only that it can be.
Examples
Example: Prove that the following numbers are rational numbers by expressing them as ratios of integers.
1. 2-4 4.
2. 64-½ 5.
3. 6. –5.4322986
4
20.3
0.96.3
Examples
Example: Prove that the following numbers are rational numbers by expressing them as ratios of integers.
1. 2-4 4.
2. 64-½ 5.
3. 6. –5.4322986
4
20.3
0.96.3
116
18
4
17
1 61
203 3
5432298610000000
Irrational Numbers
Unfortunately, the Pythagoreans themselves later discovered that the side of a square and its diagonal could not be expressed as a ratio of integers.
Prove is irrational.
Proof (by contradiction): Assume is rational. This means that there exist relatively prime integers a and b such that
2
2
2
22 2a ab b
2 22 , therefore, is evenb a a
Irrational Numbers
This means there is an integer j such that 2j=a.
If a and b are both even, then they are not relatively prime. This is a contradiction. Therefore, is irrational.
Theorem: Let n be a positive integer. Then is either an integer or it is irrational.
222 2b j2 22 4b j
2 22 is evenb j b
2
n
Real Numbers
The number line is a geometric model of the system of real numbers. Rational numbers are thus fairly easy to represent:
What about irrational numbers? Consider the following:
-2 -1 1 2
-1
1
x
y
(1,1)
2
Real Numbers
In this way, if an irrational number can be identified with a length, we can find a point on the number line corresponding to it.
What this emphasizes is that the number line is continuous—there are no gaps.
Intervals
ba
ba
ba ba
ba
ba
Name of Interval
NotationInequality
DescriptionNumber Line Representation
finite, open (a, b) a < x < b
finite, closed [a, b] a x b
finite, half-open (a, b]
[a, b)
a < x b
a x < b
infinite, open (a, )(-, b)
a < x < - < x < b
infinite, closed [a, )
[-, b]
a x < -< x b
ba
ba
ba
ba
b
a
b
a
Finite and Repeating Decimals
If a nonnegative real number x can be expressed as a finite sum of of the form
where D and each dn are nonnegative integers and 0 dn 9 for n = 1, 2, …, t, then D.d1d2…dt is the finite decimal representing x.
1 22 ...
10 10 10
t
t
d d dx D
Finite and Repeating Decimals
If the decimal representation of a rational number does not terminate, then the decimal is periodic (or repeating). The repeating string of numbers is called the period of the decimal.
It turns out that for a rational number where b > 0, the period is at most b – 1.
ab
Finite and Repeating Decimals
Example: Use long division (yes, long division) to find
the decimal representation of and find its period.
What is the period of this decimal?
46213
Finite and Repeating Decimals
Example: Use long division (yes, long division) to find
the decimal representation of and find its period.
What is the period of this decimal?
46213
46235.538461
13
6
Finite and Repeating Decimals
The repeating portion of a decimal does not necessarily start right after the decimal point. A decimal which starts repeating after the decimal point is called a simple-periodic decimal; one which starts later is called a delayed-periodic decimal.
1 2 30. ... td d d d ( 0)td
0.3, 0.142857, 0.1, 0.09, 0.0769231 2 30. ... pd d d d
0.16, 0.083, 0.0714285, 0.06 1 2 3 1 2 30. ... ...t t t t t pd d d d d d d d
Type of Decimal Examples General Form
terminating 0.5, 0.25, 0.2, 0.125, 0.0625
simple-periodic
delayed-periodic
Decimal Representation
If we know the fraction, it’s fairly straightforward (although sometimes tedious) to find its decimal representation. What about going the other direction? How do we find the fraction from the decimal, especially if it repeats?
We’ve already seen how to represent a terminating decimal as the sum of powers of ten. More generally, we can state that the decimal 0.d1d2d3…dt can be written as
, where M is the integer d1d2d3…dt.10t
M
Decimal Representation
For simple-periodic decimals, the “trick” is to turn them into fractions with the same number of 9s in the denominator as there are repeating digits and simplify:
To put this more generally, the decimal
can be written as the fraction , where M is the
integer d1d2d3…dp.
3 10.3
9 3
9 10.09
99 11
153846 20.153846
999999 13
1 2 30. ... pd d d d
10 1p
M
Decimal Representation
For delayed-periodic decimals, the process is a little more complicated. Consider the following:
What is the decimal representation of ?
is the product of what two fractions?
Notice that the decimal representation has characteristics of each factor.
112
0.083
112
1 1
4 3
Decimal Representation
It turns out you can break a delayed-periodic decimal into a product of terminating and simple-periodic decimals, so the general form is also a product of the general forms:
The decimal can be written
as the fraction , where N is the integer
d1d2d3…dtdt+1dt+2dt+3…dt+p – d1d2d3…dt .
1 2 3 1 2 30. ... ...t t t t t pd d d d d d d d
10 10 1t p
N
Decimal Representation
Example: Convert the decimal to a fraction.
0.467988654
Decimal Representation
Example: Convert the decimal to a fraction.
It’s possible this might reduce, but we can see that there are no obvious common factors (2, 3, 4, 5, 6, 8, 9, or 10), so it’s okay to leave it like this.
0.467988654
3 6
467988654 467 4679881870.467988654
99999900010 10 1